Problema Sejam [tex]\\alpha, \\beta, \\gamma[\/tex] as ra\u00edzes da equa\u00e7\u00e3o polinomial do terceiro grau [tex]x^3-x^2-2x+1=0[\/tex]. Determine o valor da express\u00e3o <\/p>\n \n<\/div>\n \n AJUDA<\/p>\n \n As Rela\u00e7\u00f5es de Girard<\/em><\/strong> para uma equa\u00e7\u00e3o c\u00fabica da forma [tex]\\boxed{x^3+kx^2+ lx+m=0}[\/tex] garantem que, se considerarmos que [tex]x_1[\/tex], [tex]x_2[\/tex] e [tex]x_3[\/tex] s\u00e3o as ra\u00edzes dessa equa\u00e7\u00e3o, ent\u00e3o: Solu\u00e7\u00e3o<\/p>\n Das rela\u00e7\u00f5es de Girard, vem:<\/p>\n [tex]\\qquad \\alpha+\\beta+\\gamma=1;[\/tex] Assim:<\/p>\n [tex]\\qquad \\sqrt[3]{\\dfrac{\\alpha^4}{\\beta^2 \\gamma^2}}+\\sqrt[3]{\\dfrac{\\beta^4}{\\alpha^2 \\gamma^2}}+\\sqrt[3]{\\dfrac{\\gamma^4}{\\alpha^2 \\beta^2}}[\/tex] Solu\u00e7\u00e3o elaborada pelos Moderadores do Blog. Problema (Indicado a partir do 3\u00ba ano do E. M.) Sejam [tex]\\alpha, \\beta, \\gamma[\/tex] as ra\u00edzes da equa\u00e7\u00e3o polinomial do terceiro grau [tex]x^3-x^2-2x+1=0[\/tex]. Determine o valor da express\u00e3o [tex]\\sqrt[3]{\\dfrac{\\alpha^4}{\\beta^2 \\gamma^2}}+\\sqrt[3]{\\dfrac{\\beta^4}{\\alpha^2 \\gamma^2}}+\\sqrt[3]{\\dfrac{\\gamma^4}{\\alpha^2 \\beta^2}}.[\/tex] AJUDA As Rela\u00e7\u00f5es de Girard para uma equa\u00e7\u00e3o c\u00fabica da forma [tex]\\boxed{x^3+kx^2+ lx+m=0}[\/tex] garantem que, se considerarmos que [tex]x_1[\/tex], [tex]x_2[\/tex] e [tex]x_3[\/tex] s\u00e3o … <\/p>\n
\n (Indicado a partir do 3\u00ba ano do E. M.)<\/span><\/p>\n
\n![\"explicador_p\"](\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/01\/menino.jpg\")
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\n[tex]\\qquad \\qquad \\textcolor{#800000}{(i)}[\/tex] [tex]x_1+x_2+x_3=-k [\/tex];
\n[tex]\\qquad \\qquad \\textcolor{#800000}{(ii)}[\/tex] [tex] x_1\\cdot x_2+ x_1\\cdot x_3+ x_2\\cdot x_3=l [\/tex];
\n[tex]\\qquad \\qquad \\textcolor{#800000}{(iiii)}[\/tex] [tex] x_1\\cdot x_2\\cdot x_3=-m[\/tex].\n<\/div>\n
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\n[tex]\\qquad \\alpha \\beta+\\beta \\gamma+ \\gamma \\alpha=-2;[\/tex]
\n[tex]\\qquad \\alpha \\beta \\gamma=-1.[\/tex]<\/p>\n
\n[tex]\\qquad =\\sqrt[3]{\\dfrac{\\alpha^6}{\\alpha^2 \\beta^2 \\gamma^2}}+\\sqrt[3]{\\dfrac{\\beta^6}{\\alpha^2 \\beta^2 \\gamma^2}}+\\sqrt[3]{\\dfrac{\\gamma^6}{\\alpha^2 \\beta^2 \\gamma^2}}[\/tex]
\n[tex]\\qquad =\\sqrt[3]{\\dfrac{\\alpha^6}{(\\alpha \\beta \\gamma)^2}}+\\sqrt[3]{\\dfrac{\\beta^6}{(\\alpha \\beta \\gamma)^2}}+\\sqrt[3]{\\dfrac{\\gamma^6}{(\\alpha \\beta \\gamma)^2}}[\/tex]
\n[tex]\\qquad =\\sqrt[3]{\\dfrac{\\alpha^6}{(-1)^2}}+\\sqrt[3]{\\dfrac{\\beta^6}{(-1)^2}}+\\sqrt[3]{\\dfrac{\\gamma^6}{(-1)^2}}[\/tex]
\n[tex]\\qquad =\\alpha^2+\\beta^2+\\gamma^2[\/tex]
\n[tex]\\qquad =(\\alpha+\\beta+\\gamma)^2-2(\\alpha \\beta+\\beta \\gamma+ \\gamma \\alpha)[\/tex]
\n[tex]\\qquad =(1)^2-2\\cdot(-2)=5.[\/tex]<\/p>\n
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\n<\/strong><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"